Multiplication Rules

Multiplication rules are very similar. Again we imagine that the reader knows how to multiply real numbers (using a calculator, even) so that the idea of multiplication of numbers as a process is familiar. In that case, the reader can easily verify for themselves that given real numbers $a$, $b$, $c$, the following are all true. They are true for complex $a$, $b$, and $c$ as well (trust me), but that's a lot easier to show after we've worked out the rules for real numbers, in particular distributivity. We have to work a bit harder on inversion and division, as well.

$$a*(b*c) = (a*b)*c \quad {\rm associativity}$$ (3.16)

$$a*b = b*a \quad {\rm commutativity}$$ (3.17)

$$a*1 = a \quad {\rm identity}$$ (3.18)

$$a*\frac{1}{a} = a*a^{-1} = 1 \quad {\rm inverse}$$ (3.19)

Once again we see that for expressions that contain only multiplied terms we can group them in any order:

$$1*(3*4) = 1*12 = 12 = 3*4 = (1*3)*4 = 1*3*4$$ (3.20)

(again, try it for different numbers). No matter what order we multiply out (symbols for) numbers, we get the same result.

There are more rules

In physics one can multiply symbols with different units, such an equation with (net) units of meters times symbols given in seconds. In the end, however, the units on both sides must be consistent and make sense.